Zagier duality for harmonic weak Maass forms of integral weight

We show the existence of"Zagier duality"between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight,"Zagier duality"between scalar valued ones is derived.


Introduction and statement of results
For an integer k, let M !+  On the other hand, for D > 0 with D ≡ 0, 1 mod 4 there is also a unique modular form g D ∈ M !+ 3 2 having a Fourier expansion of the form As proven by Zagier [21, Theorem 1] the g 1 (τ ) is essentially the generating function for the traces of singular moduli. He also proved the so-called "Zagier duality" [21,Theorem 4] relating the Fourier coefficients of f d (τ ) and g D (τ ): Zagier's results have inspired vast research subjects (for instance see [3,12,13,15,16,17,18,20]) and have been extended to study duality properties on the space of harmonic weak Maass forms H k (Γ 0 (N)). For instance, in terms of the weight (higher weight) and space (weak Maass forms) aspects, K. Bringmann and K. Ono [3,Theorem 1.1] showed Zagier duality between certain Maass-Poincaré series of weight −k + 3/2 and Poincaré series of weight k + 1/2 for Γ 0 (4) with k ≥ 1. In fact, their Fourier coefficients are traces of singular moduli of certain weight 0 Maass forms. A. Folsom and K. Ono [15] found Zagier duality between certain weight 1/2 harmonic weak Maass forms and weight 3/2 weakly holomorphic modular forms on Γ 0 (144) with Nebentypus ( 12 · ). The holomorphic part of their initial Maass form is essentially Ramanujan's mock theta function f (q). Also, C. H. Kim [18] proved Zagier duality between certain weakly holomorphic modular forms of weight 1/2 and 3/2 on Γ 0 (4p).
Here p is a prime such that the genus of the Fricke group Γ 0 (p) * equals 0. In his result, those Fourier coefficients are essentially traces of singular moduli of the Hauptmodul j * p (τ ) for Γ 0 (p) * . Now we introduce the results of integral weight cases. J. Rouse [20,Theorem 1] proved Zagier duality between certain weakly holomorphic modular forms of weight 0 and 2 on Γ 0 (p) with Nebentypus ( · p ) where p = 5, 13, 17. D. Choi [11, Theorem 1.2] gave a simple proof of Rouse's result by using the residue theorem, and extended to any odd prime level p. Note that their results concern weakly holomorphic modular forms. Recently P. Guerzhoy [16, Theorem 1] showed that there is Zagier duality between certain harmonic weak Maass forms of weight k and weakly holomorphic modular forms of weight 2 − k for SL 2 (Z) where k ≤ 0 is even.
The purpose of this paper is to derive Zagier duality for harmonic weak Maass forms and, as a result, this extends the previously known Zagier duality between integral weight forms. To this end, we first show that duality holds between the space H k,ρ L of vector valued harmonic weak Maass forms and the space M ! 2−k,ρ L of vector valued weakly holomorphic modular forms with k ≤ 0 an integer (Theorem 1.2). This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades [8,9]. By taking L in our result as an even unimodular lattice, we immediately recover the recent result by Guerzhoy [16] (Corollary 1.3). Now we take L so that a certain space H ǫ k (Γ 0 (p), ( · p )) of scalar valued harmonic weak Maass forms is isomorphic to H k,ρ L (Proposition 1.4). Then, as another corollary, for any odd prime p we have Zagier duality between H ǫ k (Γ 0 (p), ( · p )) and a certain space M !δ 2−k (Γ 0 (p), ( · p )) of scalar valued weakly holomorphic modular forms (Corollary 1.5).
To state our main theorem let L be a non-degenerate even lattice of signature (b + , b − ), and L ′ its dual lattice. We denote the standard basis elements of the group algebra C[L ′ /L] by e γ for γ ∈ L ′ /L. Let ρ L be the Weil representation associated to the discriminant form (L ′ /L, Q), andρ L its dual representation. We write M ! k,ρ L for the space of C[L ′ /L]-valued weakly holomorphic modular forms of weight k and type ρ L , and H k,ρ L for the space of C[L ′ /L]-valued harmonic weak Maass forms of weight k and type ρ L (see Section 2). In the following theorem we obtain Zagier duality between the space H k,ρ L of vector valued harmonic weak Maass forms and the space M ! 2−k,ρ L of vector valued weakly holomorphic modular forms with k ≤ 0 an integer.
If we take L in Theorem 1.2 as a unimodular one, we can recover Guerzhoy's result [16, Theorem 1].
If k < 0, then f m , g n are uniquely determined.
Of course we may take L as another suitable lattice. Let p be an odd prime. We write ) for the space of weakly holomorphic modular forms of integral weight k for Γ 0 (p) with Nebentypus ( · p ). For ǫ ∈ {±1} we define the subspace Suppose that the discriminant group L ′ /L is isomorphic to Z/pZ. Then b + − b − is even, and the quadratic form on L ′ /L is equivalent to Q(γ) = λγ 2 p for λ, γ ∈ Z/pZ with λ = 0. Now we put ǫ = ( λ p ), δ = ( −1 p )ǫ, and assume that k ≡ (b + − b − )/2 mod 2. Then Bruinier and Bundschuh [7,Theorem 5] showed that the space M !ǫ k (Γ 0 (p), ( · p )) (resp. M !δ k (Γ 0 (p), ( · p ))) of scalar valued weakly holomorphic modular forms is isomorphic to the space M ! k,ρ L (resp. M ! k,ρ L ) of vector valued ones. We observe that Bruinier and Bundschuh's argument can also be applied to the spaces of scalar and vector valued harmonic weak Maass forms. Let H k (Γ 0 (p), ( · p )) denote the space of harmonic weak Maass forms of weight k for Γ 0 (p) with Nebentypus ( · p ) (see Section 2).
Here Γ(a, y) = ∞ y e −t t a−1 dt denotes the incomplete Gamma function. For ǫ ∈ {±1} we define the subspace For a given Remark 2. A similar result for half integral weight case can be found in [10].
Corollary 1.5. With the notation and assumption as in Proposition 1.4, we further assume that k ≤ 0. For m, n ∈ Z >0 with ( −m p ) = −ǫ and ( −n p ) = −δ, there exist f m ∈ H ǫ k (Γ 0 (p), ( · p )) and g n ∈ M !δ 2−k (Γ 0 (p), ( · p )) with Fourier expansions of the form If k < 0, then f m , g n are uniquely determined. Recall that weakly holomorphic modular forms of weight k for Γ 0 (N) with Nebentypus χ are holomorphic functions f : H → C which satisfy: (ii) f has a Fourier expansion of the form and analogous conditions are required at all cusps. Here q = e 2πiτ as usual.
We write M ! k (Γ 0 (N), χ) for the space of these weakly holomorphic modular forms. A smooth function f : H → C is called a harmonic weak Maass form of weight k for Γ 0 (N) with Nebentypus χ if it satisfies: (ii) ∆ k f = 0, where ∆ k is the weight k hyperbolic Laplace operator defined by We denote the space of these harmonic weak Maass forms by H k (Γ 0 (N), χ). This space can be denoted by H + k (Γ 0 (N), χ) in the context of [8]. Here we follow notation given in [9]. Then the space H k (Γ 0 (N), χ) contains the space M ! k (Γ 0 (N), χ) of weakly holomorphic modular forms of weight k for Γ 0 (N) with Nebentypus χ. The polynomial P f ∈ C[q −1 ] is called the principal part of f at the corresponding cusps. In particular f ∈ H k (Γ 0 (N), χ) has a unique decomposition

2.2.
Vector valued modular forms. We write Mp 2 (R) for the metaplectic two-fold cover of SL 2 (R). The elements are pairs (M, φ), where M = ( a b c d ) ∈ SL 2 (R) and φ : H → C is a holomorphic function with φ(τ ) 2 = cτ + d. The multiplication is defined by We denote byρ L the dual representation of ρ L .
Let k ∈ 1 2 Z. A holomorphic function f : H → C[L ′ /L] is called a weakly holomorphic modular form of weight k and type ρ L for the group Mp 2 (Z) if it satisfies: (ii) f is meromorphic at the cusp ∞.
Here condition (ii) means that f has a Fourier expansion of the form The space of these C[L ′ /L]-valued weakly holomorphic modular forms is denoted by M ! k,ρ L . Similarly we can define the space M ! k,ρ L of C[L ′ /L]-valued weakly holomorphic modular forms of typeρ L . The subspace of C[L ′ /L]-valued holomorphic modular forms (resp. cusp forms) of weight k and type ρ L is denoted by M k,ρ L (resp. S k,ρ L ).
A smooth function f : H → C[L ′ /L] is called a harmonic weak Maass form of weight k and type ρ L for the group Mp 2 (Z) if it satisfies: We denote by H k,ρ L the space of these C[L ′ /L]-valued harmonic weak Maass forms. This

2.3.
Zagier duality for weakly holomorphic modular forms. We begin with the following rather simple observation.
Proof. Since γ∈L ′ /L f γ (τ )g γ (τ ) is holomorphic on H and meromorphic at the cusp ∞, it suffices to verify the modular transformation property. First note that Here ·, · denotes the usual dot product.

Now we find for
Since ρ L is unitary we have Then γ∈L ′ /L m+n=0 c f (γ, m)c g (γ, n) = 0.
Proof. By Lemma 2.1, γ∈L ′ /L f γ g γ is a weakly holomorphic elliptic modular form of weight 2. Thus by the residue theorem we immediately obtain the assertion.  3. The results of Bruinier, Funke, Ono, and Rhoades [8,9] Assume that k ≤ 0 is an integer. Recall that there is an antilinear differential operator where L k := −2iy 2 ∂ ∂τ is the Maass lowering operator (see [8,9]). The Maass raising operator is defined by R k := 2i ∂ ∂τ + ky −1 . By [8,Corollary 3.8] the following sequence is exact: Using the Petersson scalar product, we can obtain a bilinear pairing between M 2−k,ρ L and H k,ρ L defined by where g ∈ M 2−k,ρ L and f ∈ H k,ρ L .
Recall the differential operator From Bol's identity [9, Lemma 2.1] one finds Proposition 3.4. If h ∈ H k,ρ L , then Moreover, we have Proof. Note that R 1−k k h satisfies the transformation behavior for vector valued modular forms of weight 2 − k and type ρ L . The remaining assertions can be derived by the same argument as in the proof of [9, Theorem 1.1].
Proposition 3.5. If f ∈ H k,ρ L and h ∈ H k,ρ L , then Proof. Note that ξ k (f ) ∈ S 2−k,ρ L is a cusp form. Now the assertion follows from [9, Corollary 4.2].
Combining Propositions 3.3-3.5 we have the following.
Theorem 3.6. Let k ≤ 0 be an integer, f ∈ H k,ρ L , and g ∈ D 1−k (H k,ρ L ) ⊂ M ! 2−k,ρ L . Then By the same argument as in [7] one can see that F satisfies the desired modular transformation property. To check ∆ k F = 0 we recall from [7, Proposition 2] that where W p := ( 0 −1 p 0 ) is the Fricke involution. Since ∆ k commutes with the Petersson slash operator (see [19]), we have ∆ k F = 0. For the converse we recall from [7, Lemma 1] that the inverse isomorphism F → f is given by Hence ∆ k f vanishes.